Analysis of employee stock options and guaranteed withdrawal benefits for life

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Analysis of Employee Stock Options and

Guaranteed Withdrawal Benefits for Life

by

Premal Shah

B. Tech. & M. Tech., Electrical Engineering (2003),

Indian Institute of Technology (IIT) Bombay, Mumbai

Submitted to the Sloan School of Management

in partial fulfillment of the requirements for the degree of

Doctor of Philosophy in Operations Research

at the

MASSACHUSETTS INSTITUTE OF TECHNOLOGY

September 2008

c

Massachusetts Institute of Technology 2008. All rights reserved.

Author . . . .

Sloan School of Management

July 03, 2008

Certified by . . . .

Dimitris J. Bertsimas

Boeing Professor of Operations Research

Thesis Supervisor

Accepted by . . . .

Cynthia Barnhart

Professor

Co-director, Operations Research Center

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Analysis of Employee Stock Options and Guaranteed

Withdrawal Benefits for Life

by

Premal Shah

B. Tech. & M. Tech., Electrical Engineering (2003),

Indian Institute of Technology (IIT) Bombay, Mumbai

Submitted to the Sloan School of Management on July 03, 2008, in partial fulfillment of the

requirements for the degree of

Doctor of Philosophy in Operations Research

Abstract

In this thesis we study three problems related to financial modeling.

First, we study the problem of pricing Employee Stock Options (ESOs) from the point of view of the issuing company. Since an employee cannot trade or effectively hedge ESOs, she exercises them to maximize a subjective criterion of value. Modeling this exercise behavior is key to pricing ESOs. We argue that ESO exercises should not be modeled on a one by one basis, as is commonly done, but at a portfolio level because exercises related to different ESOs that an employee holds would be coupled. Using utility based models we also show that such coupled exercise behavior leads to lower average ESO costs for the commonly used utility functions such as power and exponential utilities. Unfortunately, utility based models do not lead to tractable solutions for finding costs associated with ESOs. We propose a new risk management based approach to model exercise behavior based on mean-variance portfolio maximization. The resulting exercise behavior is both intuitive and leads to a computationally tractable model for finding ESO exercises and pricing ESOs as a portfolio. We also study a special variant of this risk-management based exercise model, which leads to a decoupling of the ESO exercises and then obtain analytical bounds on the implied cost of an ESO for the employer in this case.

Next, we study Guaranteed Withdrawal Benefits (GWB) for life, a recent and popular product that many insurance companies have offered for retirement plan-ning. The GWB feature promises to the investor increasing withdrawals over her lifetime and is an exotic option that bears financial and mortality related risks for the insurance company. We first analyze a continuous time version of this product in a Black Scholes economy with simplifying assumptions on population mortality and obtain an analytical solution for the product value. This simple analysis reveals the high sensitivity the product bears to several risk factors. We then further in-vestigate the pricing of GWB in a more realistic setting using different asset pricing models, including those that allow the interest rates and the volatility of returns to be

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stochastic. Our analysis reveals that 1) GWB has insufficient price discrimination and is susceptible to adverse selection and 2) valuations can vary substantially depending on which class of models is used for accounting. We believe that the ambiguity in value and the presence of significant risks, which can be challenging to hedge, should create concerns to the GWB underwriters, their clients as well as the regulators.

Finally, many problems in finance are Sequential Decision Problems (SDPs) under uncertainty. We find that SDP formulations using commonly used financial metrics or acceptability criteria can lead to dynamically inconsistent strategies. We study the link between objective functions used in SDPs, dynamic consistency and dynamic programming. We then propose ways to create dynamically consistent formulations. Thesis Supervisor: Dimitris J. Bertsimas

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Acknowledgments

First, I would like to express my sincere gratitude towards Prof. Dimitris Bertsimas for his guidance and support in the work related to this thesis and beyond. From day one of my association with him, Prof. Bertsimas was never short of interesting research ideas and insights into problems. His emphasis on the practicability of research has kept me true. Without him this work would have been but an exercise in futility.

Dimitris has been a great source of inspiration, encouragement and counsel to me outside research and academics as well. His versatility in terms of dealing with a broad range of research areas in both theory and applications, his repertoire of interests and activities and his drive are stupendous. I consider myself very fortunate to have been able to tap into the same, especially given the several sources competing for his time!

I would also like to acknowledge the members of my thesis committee - Prof. Jiang Wang, for valuable inputs and suggestions to improve this thesis and Prof. David Gamarnik, for his encouragement and interest. Discussions with David about any problem have always refined my understanding significantly. I am also thankful to David, Dimitris and Prof. John Tsitsiklis for giving me the opportunity to serve as a teaching assistant with them. Teaching a course with one of them at charge has been a surprisingly good learning experience as well!

I would also like to express my gratitude towards the Operations Research Center administrators - Paulette Mosley, Laura Rose and Andrew Carvalho - their efficiency and co-operation have made things so much easier.

The years at ORC were sweet and memorable thanks to the fellow ORCers - Ilan Lobel, Yunqing (Kelly) Ye, Vinh Doan, Ruben Lobel, Doug Fearing, Dan Iancu, Carine Simon, Gareth, Juliane, Dung, Mike, Rajiv, Pranava, Shobhit, Carol, Pavithra, Theo, Andy, Yann, Elodie, Katy, Rags, Pallav, Dmitriy, Dave, Nikos, Margret, Lincoln, Karima, Mallory, Shubham, Thibault, Martin, Shashi, Sun Wei, Kostas, Nelson, ... . Thank you for the countless wonderful moments inside and outside the ORC. You all will be sorely missed!

Thanks is also due to all the room-mates and their extensions - Sudhir Chiluveru, Saurabh Tejwani, Hoda, Jean-Paul, Alec, Kurt, Raf, Mukul, Riccardo, Anatoly who made leaving in the dorms so much more homely.

I am also grateful to MIT for the opportunity to meet and make so many won-derful friends - Rajiv Menjoge, Aditya Undurti, Theta Aye, Vikas Sharma, Kranthi

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Vistakula, Neha Soni, Mithila Azad, Mira Chokshi. Thanks to them and the rest of the kajra-re and wandering gopis gangs - Jen Fang, Sukant, Preet, Tulika, Silpa, Natasha, Tara(2!), Nick, Silpa, Saba(2!), Kartik, Chief, ... for all the fun times!

The time spent here, in the cold of Boston, also helped me to thaw some old friendships - Nandan Bhat, Ajay Raghavan, Suchit Kaul and Prafulla Krishna - it was great to have rediscovered you all!

I am also thankful to Sangam and the ’05 executive council fellow members -Vivek Jaiswal, Saurabh, Biraja for helping to re-create here Diwali, Holi and all the other things that I missed from home. I would also like to thank the members of the Edgerton Board ’07 - Yvonne, Nicholas, Kelly, Biz, Grace, Becky, Ron,... and the Edgerton House Masters - David and Pamela Mindell; working with you all was a great experience.

The last several months at MIT have been different and not just because of this thesis. Thanks to Tara Sainath for helping with proof-reading this thesis and a million other things and all the care and affection I could have asked for.

Finally and most importantly, I would like to thank my parents, my sister Payal and my relatives in India - their love, support and care have been a source of great strength for me. Nothing would have been possible without this foundation of my life.

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7 Summary and Closing Remarks 183 A Relative Order of ESO Exercises 187 B Supplementary Results for Chapter 3 189 C Illustration of GWB Product Evolution 195 D Whittaker Functions and some basic properties 199 E Joint distribution of Bt and Ct 203 F Mortality Table 207 G Computational Methods for Chapter 5 209 G.1 BSM Model . . . 209

G.2 SILN Model . . . 211

G.3 SISV Model . . . 214

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List of Figures

2-1 Employee’s belief about stock price dynamics. Numbers on arrows indicate a probability of transition while the circles enclose realized stock price. . . 49 3-1 Variation of Delta-Barrier function w.r.t. moneyness and time to

ma-turity. . . 73 3-2 Exercise multiple v/s and time to expiry for different values of ν. . . 77 4-1 Average Residual life as a function of Age . . . 113 4-2 Break Even fee as a function of Age A for different asset mix choices

under the CBSME Model. . . 114 4-3 Net Value as a function of Age for different asset mix choices under

the CBSME Model. . . 115 4-4 Break even fee for the meta-portfolio as a function of risk-free rate

under the CBSME Model. . . 116 4-5 Net Value of the meta-portfolio as a function of risk-free rate for the

CBSME Model. . . 117 5-1 Break even fee for the meta portfolio as a function of risk-free rate for

the BSM model. . . 140 5-2 Net value of the meta portfolio as a function of risk-free rate for the

BSM model. . . 143 5-3 Break even fee as a function of Cohort Age for different models and

asset mixes assuming contract compliant withdrawals. . . 144 5-4 Net value as a function of Cohort Age for different models and asset

mixes assuming contract compliant withdrawals. . . 145 5-5 Variation of hedging exposure with time for delta hedging under the

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6-1 Illustration of Co-ordination between DMs for Dynamic Consistency . 169 C-1 Evolution of the GWB state variables with time for the example in

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List of Tables

3.1 Cost Estimates (Upper bounds and Lower bounds) for an ATM ESO for different values of the exercise threshold. . . 83 4.1 Distribution statistics of GWB for life net value in different scenarios

for an account started between Dec. 1949 and Dec. 1972 (for initial investment of 100,000). . . 99 4.2 Distribution statistics of GWB for life payouts in different scenarios

for an account started between Jan. 1871 and Dec. 1972 (for initial investment of 100,000). . . 99 4.3 Distribution statistics of GWB for life net value in different scenarios

for an account started between Jan. 1871 and Dec. 1972 (for initial investment of 100,000). . . 100 4.4 Maximum Phase 1 years and Average Phase 2 years for select cohorts. 110 4.5 Liability Values for different cohorts and asset mixes under the CBSME

Model. . . 111 4.6 Revenue Values/Fees for different cohorts and asset mixes under the

CBSME Model. . . 112 4.7 Break-even fees for select cohorts and asset mixes under the CBSME

Model. . . 112 4.8 Net value of the GWB product for select cohorts and asset mixes under

the CBSME Model. . . 112 4.9 Impact of changes in average withdrawal years on break-even fee and

net value of the meta-portfolio under the CBSME Model. . . 116 5.1 Break-even fees under different models. . . 138 5.2 Net value under different models. . . 139 5.3 Value of insurance company liabilities from GWB for life under

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5.4 Value of revenues from GWB for life under different models. . . 142 5.5 Sensitivity of the meta portfolio value to mortality rates under different

models assuming contract compliant withdrawals. . . 146 C.1 Numerical illustration of evolution of the GWB product related

vari-ables and cash-flows in a hypothetical scenario. . . 197 F.1 ERISA Section 4050 mortality rates for ages 49 and above (year 2008).

Average residual life for each age is computed using the De-Moivre’s approximation. . . 207

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Chapter 1 Introduction

1.1 Motivation

Much of quantitative finance is based on models of market variates - how prices of securities traded in the markets evolve, how market participants behave and the interaction between the two.

In a landmark paper, Black and Scholes [18] proposed a model to price a stock-option based on the price of the underlying stock. They provided not only a way to unambiguously price an option, but also a method to “hedge” out and in theory, eliminate the risk of holding or underwriting this option. This spawned an entire new field of financial engineering and a fresh body of work based on the concept of risk-neutral pricing and hedging was created and is still being pursued very actively. Simultaneously, in industry, a plethora of complex financial derivatives are being created, marketed and sold to institutions and households.

The assumptions underlying the risk neutral pricing theory are that the markets are complete and that the market players do not face constraints in buying and sell-ing the various instruments traded in the market. In this ideal settsell-ing, all derivative securities are redundant and can not only be priced unambiguously but also hedged perfectly from the prices of other traded instruments. The quantitative link between various traded instruments in the market is established by using a parametric model for price process of the underlying(s) on which the derivative instruments are writ-ten. This method works fairly well for pricing standard instruments that are heavily traded. But as the derivative instrument to be priced becomes more intricate in its dependence on the underlying(s) and the market for it more constrained, the finan-cial engineering methodology becomes less precise and more subjective. The model

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that is used to link together prices of different securities then itself becomes a key factor in valuation. Due care must then be taken in creating models to price such an instrument as well as interpreting results from the so called no-arbitrage pricing models.

In this thesis, we examine two problems in pricing derivatives in “incomplete” markets and show how even reasonable models can sometimes leave out significant determinants of value.

Models for behavior of market participants such as investment managers or con-sumers are in general decision and control problems. Again here, we observe in a dynamic setting, an ad-hoc formulation of an investment manager’s problem that employs the commonly used financial metrics can lead to a model where the man-ager takes conflicting decisions over time. We also examine in this thesis, general properties of a dynamic decision framework for “consistency” in decisions.

1.2 Contributions

Modeling Exercise Behavior for pricing Employee Stock

Op-tions

We consider the problem of pricing Employee Stock Options (ESOs) from the point of view of the companies that issue them. Although, off late, ESOs have been losing popularity due to scandals and less favorable accounting regulations, they still con-stitute a sizeable chunk of many companies’ compensation costs. Since employees are constrained in both trading as well hedging ESOs, the standard risk-neutral pricing framework cannot be directly applied to ESOs. Employees, would exercise ESOs to maximize a personal measure of value or utility. Pricing ESOs then encompasses twin problems - we first need to model employee exercises, which can be sub-optimal from a risk neutral perspective, while being subjectively optimal; and then compute the risk-neutral cost of the ESOs under these exercise policies. The choice of model for exercise behavior, will have a big impact on the valuation of ESO costs.

Most ESO exercises are driven by the need of a risk-averse employee to limit the uncertainty of an option payoff that she cannot hedge. The basic financial tenet of diversification would suggest that as all ESOs are exposed to common risk sources, they have diminishing marginal value to the employee. It is then plausible that incremental option grants will, in general, be exercised differently. This, in turn, would lead to a different incremental cost for an ESO grant to the employer. The

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models that have been proposed in literature, typically price ESOs on an individual basis and in isolation and would not take this effect into consideration. We propose a new approach - to explicitly take into account the employee’s need for diversification with respect to the entire portfolio of ESOs that she holds while modeling her exercise behavior. Thus we propose to price ESOs at a portfolio level rather than at individual level.

We first augment the conventional utility based framework that leads to an en-dogenous exercise model for ESOs and show that in general bundling together of ESOs affects exercise behavior and tends to cause an employee to exercise her ESOs earlier on average. This makes her forgo a larger part of the option value of the port-folio, thereby reducing its cost for the employer. Also, an immediate consequence of taking these portfolio effects into account is that the cost of an option portfolio is no longer linear or equal to the sum of its parts. Further, issuance of new options can have a retrospective effect on the cost of already issued options.

We then use the concept of risk management and ideas in portfolio optimization to motivate a model, where the employee exercises options so as to optimize a risk-adjusted value of the entire portfolio at each time step. This causes the employee to exercise options in decreasing order of a barrier function that can be interpreted as a pseudo Sharpe-ratio for the option. The advantage of this model is that it leads to a computationally simple framework to both price the ESO portfolio and also allocate its costs amongst its components. For a special risk-management based exercise model we show that option exercises decouple and one can think of applying the “pseudo Sharpe-ratio” criterion to options on a one-by-one basis. In this case, we recover a linear pricing rule for ESOs and also derive tight analytical bounds on the cost of an ESO.

Pricing Guaranteed Withdrawal Benefits for Life

Complex financial derivatives are often embedded in retail investment products. We consider one such recent and extremely popular innovation in the private pension product space - the Guaranteed Withdrawal Benefits (GWB) for life. The GWB for life option, which is usually available as an add-on to a Variable Annuity (VA) investment fund, guarantees an investor a non-decreasing stream of payments in her retirement until death, with her funds always staying invested in the VA. In return, the investor pays a small fee indexed to the quantum of the guarantee, every year. While prima-facie, GWB for life appears to be just another, somewhat exotic, financial

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option, pricing it poses many challenges. Due to its exposure to population longevities and dependence on investor behavior over time and complex dependence on various financial market factors, the complete markets hypothesis does not hold for the GWB. We first undertake an analysis of GWB for life in a simplified setting using the Black-Scholes model for asset prices. This allows us to get an almost closed form solution for the value of GWB. We use this solution to draw insights and investigate the impact of potential risk factors and find almost all of them to be quite significant. We then price GWB in a more realistic setting using models that allow interest rates and equity-market volatilities to be stochastic. We find that accounting for these additional risks can alter valuations significantly. In addition, GWB has considerable exposure to realized investor population longevity. These facts suggest that hedging GWB is likely to be only partially successful in practice. We also find that the typical GWB for life offering with its uniform pricing across fund classes and investor ages is susceptible to adverse selection in its customer profile and needs price discrimination.

Dynamic Consistency For Sequential Decision Problems

Portfolio Optimization and Risk Management are standard problems in finance. In a dynamic setting, these can be viewed as instances of a much broader class of problems - Sequential Decision Problem (SDP)s. We study SDPs in context of an important normative criterion for a good SDP model - that it should lead to dynamically con-sistent planning. A lack of dynamic consistency would mean that the decision maker would make plans, while being fully aware that she will not carry them through. While SDPs arise in several decision and control settings, we show that tendency to be dynamically inconsistent is particularly severe for financial applications. This is because SDPs based on many of the performance metrics such as Sharpe-ratio, vari-ance adjusted mean, Value at Risk etc as well as acceptability criteria based on so called dynamic risk measures turn out to be dynamically inconsistent.

We explore the connection between dynamic consistency and the algorithmic no-tion of dynamic programming or Bellman’s principle and find them to be closely related, though not identical. We show that most dynamically consistent strategies can be considered to be arising from SDPs that have a sum decomposable represen-tation across time and event space. We then examine how “conflicts” due to dynamic inconsistency can be resolved for specific applications. We also propose a new class of dynamically consistent performance metrics that are essentially expectations with respect to probability measures distorted in a specific way.

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1.3 Thesis Overview

The thesis is organized as follows. In Chapter 2, we introduce the problem of ESO pricing for the company that issues them. We examine how the presence of other ESOs in an employee’s portfolio can affect the exercise decisions concerning ESOs and thereby their cost to the company. In Chapter 3, we then propose risk management based models for exercise behavior, and through them, a tractable way to price ESO portfolios. In Chapter 4, we then turn to the problem of pricing GWB for life. In this chapter, we propose an analytical framework to price GWB for life for a continuous time counterpart of the product. In Chapter 5, we price GWB for life using realistic models and investigate the impact of modeling interest rates and volatilities as stochastic processes on pricing. In Chapter 6, we look into the issue of dynamic consistency for SDPs, especially in the context of some standard problems in finance. We summarize the findings and some interesting research directions in Chapter 7.

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Chapter 2 Pricing Employee Stock Options

2.1 Introduction

Employee Stock Options (ESOs) are commonly used by corporations as an effective, but often controversial, form of compensation for mid and high level employees. An ESO is typically an American Call Option that the employee can exercise between two pre-specified dates, the earlier date called the Vesting Date and the later the Expiration Date. By its very structure, an ESO acts like a performance - linked compensation for higher echelon executives. Moreover, the employing company would realize the cost of this pay only in the event of its stock performing well.

The popularity of ESOs as a means of compensation for employers can be at-tributed to three important reasons:

• They directly link pay to performance and serve to align, at least partially, the management’s interests with those of the shareholders, thereby mitigating some of the “agency problems”1.

• Long-term options create an incentive for the employee to stay with the company and thus options with vesting schedules can be used to retain talent.

• Most ESOs are at the money (ATM) call options. Until recently, under Fi-nancial Accounting Standards Board (FASB)’s alternate accounting provisions, companies could expense stock option grants to employees at their intrinsic value, i.e., zero costs for the ATM options. Thus this form of compensation,

1_{Agency problems arise because in general the goals and objectives of a company’s management}

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at grant, would not add to company’s expenses or show up in its profit and loss/earnings statements.

On the negative side, ESOs issued to executives can also create a conflict of in-terest between management and shareholders, as the asymmetric option payoff can incentivize managers to undertake projects with unduly high risk. More important, perhaps, is the fact that ESOs actually amount to a significant liability on a com-pany’s balance-sheet that often goes under-expensed. For example, when ESOs issued by technology companies were exercised by the employees during the dot-com boom, there were payoffs amounting to tens and sometimes hundreds of millions of dollars. These were effectively a transfer of value from the shareholders to the employees, mainly executives.

2.1.1 Motivation

Empirical surveys show high-level executives receive a bulk of their compensation as stock options, a trend which has recently started to somewhat reverse because of scandals and controversies. For example, according to the data compiled by Hender-son [67], in 2002, 58% of the net CEO pay in the US and 24% in the UK was options related. In terms of balance-sheet liabilities, Hall and Murphy, [63] report that in 1992, firms in the Standard & Poor’s 500 granted their employees options worth a total of 1.1 billion at the time of grant. This figure reached 119 billion in 2000 before dropping down to 71 Billion in 2002, still a sizeable figure. Because ESO related costs can amount to a substantial fraction of the firms’s balance-sheet, evaluating this cost is important for investors and regulators.

Pricing ESOs is however made difficult because of the fact that they are not tradeable and hence do not have a directly observable market price. We also cannot price these options using standard models such as the Black-Scholes framework [18], because the option bearer faces constraints that would not allow her to hedge these options effectively2. This coupled with risk-averseness causes a typical employee to exercise an ESO in a way that would substantially reduce its cost below its Black-Scholes or risk-neutral price.

More specifically, the employing company realizes the ESO cost if and when the employee exercises the option. Unlike a regular call option which typically gets

ex-2_{To hedge a call option, the employee should be able to short the employing company’s stock.}

This is usually prohibited by regulatory bodies. Also if the employee were able to hedge out the options, then many of the objectives of issuing the options as a means of incentive and retentive compensation will be lost.

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ercised at or close to expiry3_{, an ESO is typically exercised much earlier, see Hall}

and Murphy [63]. ESOs follow subjective exercise patterns that are difficult to pre-dict because the employee exercise ESOs to realize a measure of personal value. The clauses related to vesting and forfeiture of options (in the event of employee quitting or being terminated) further complicate the problem of pricing ESOs.

Unfortunately, but perhaps also unsurprisingly, there is no consensus in literature or in practice about what the fair cost of ESOs is. After some accounting contro-versies and several debates, the FASB issued a revision to Statement 123 that deals with accounting principles for stock related compensation in 2004 [54]. This made it mandatory for publicly traded corporations to expense ESOs at their “fair value” (or levels more representative of the cost incurred than the intrinsic value accounting), effective 2005. The European counterpart, International Accounting Standards Board (IASB), had laid down similar stipulations earlier through [74]. However FASB, IASB and other regulatory bodies have only laid down broad guidelines when it comes to methods and models to estimate a “fair cost” of ESOs, going only so far as indicating preferences for some models - such as the lattice model. For example, the Securities and Exchange Commission (SEC) Bulletin guidelines [108], state that the accounting practice used to price an ESO must be based on sound financial economic theory and be generally accepted in the field but stops short of laying down a specific accounting rule, see Cvitanic and Zapatero [47]. While it is broadly agreed that the true cost of an ESO lies between its intrinsic value and the Black-Scholes value, there continues to be an active debate about what exactly the “fair cost” of an ESO is.

In this chapter, and Chapter 3, we seek to develop a framework to model ESO exercises so as to estimate the cost of the outstanding ESOs on a company’s balance-sheet. Our focus in this chapter will be to understand the functional nature of ESO costs. In Chapter 3, we seek practical methods to expense ESOs that are driven by economic reasoning and at the same time are simple to implement in practice.

While not considered in this thesis, an interesting problem related to costing ESOs is estimating the value of ESOs to the employee. This has been studied well e.g. Lambert, Larcker and Verrechia in [83], Ingersoll in [73]. Understanding how an employee would value the options grant is useful in designing compensation packages to incentivize desired management or employee actions. How an employee would value an option grant though will not be the same as the cost it represents to the company. In this context, it is worth mentioning a generally accepted conclusion that

3_{It is well-known that an American Call Option on the stock of a company that does not pay}

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the option’s value to the employee is less than the cost of issue to the employer as has been discussed by Hall and Murphy in [62] and [63]. The difference, a “deadweight loss”, may also be seen as a price that the company pays to solve its agency problems and retain talent, see Kadam [77].

2.1.2 Related Work

ESOs have been an active topic of research and debate and there is a vast body of work that deals with pricing of ESOs and other issues related to them. As the topic touches so many fields, the contributions also come from diverse areas including Accounting, Econometrics, Asset Pricing and Mathematical Finance. Chance [32] provides a de-tailed analysis of the issues related to ESOs from many different perspectives as well as a sound critique of the approaches that have been proposed to address these issues in the literature. Hall and Murphy [63] analyze historical trends in issuance of ESOs by corporations to executives and lower level employees and the possible attractions and pitfalls of using them as incentive - pay. Another paper by the same authors, [62], provides a good understanding of the role of ESOs in incentivizing executives and how risk-averseness and other idiosyncratic investor characteristics might affect exercise behavior using a stylized model. The authors propose a simple utility based framework and give numerical examples to illustrate the effects of risk-averseness and trading restrictions on employee’s exercise behavior and cost of ESOs to the issuers. Huddart and Lang [71] present an empirical analysis of how employees tend to ex-ercise their ESOs using over 10 years of data. Bettis, Bizjak and Lemmon [15] provide an analysis of exercise behavior and incentive effects of ESOs using an empirically calibrated utility model.

The employee’s decision-process remains fundamental to pricing an ESO4_{. Hence,}

even though we do not seek to value ESOs from an employee’s perspective, we still need to have a model of the employee’s exercise behavior.

In general, for modeling exercise behavior, two broad approaches have been used, as observed in Carr and Linetsky [31]. The first approach treats ESO exercise as an endogenous process and models it as a decision triggering from typically a utility optimization consideration. There are several factors that can potentially influence

4_{However, approaches to pricing, that circumvent this have also been proposed. For example,}

Bulow and Shoven [27] suggest an alternate way of accounting for ESO costs, in which the ESOs are expensed as rolling options of quarterly maturity until they are exercised or lapse. In another strikingly different approach to pricing, Core and Guay [43] suggest an empirically calibrated model that uses data available in a company’s proxy statement to price ESOs.

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exercise decisions:

• Employee’s Risk Averseness - Employees are typically over-exposed to their employers. The option-bearer might want to offload some of this by exposure by cashing the ESOs.

• Tax Implications.

• Career related moves can cause early ESO exercises or forfeitures. • Liquidity crunches may force an early exercise of the option.

• ESO terms and company policies - Companies tend to reset strikes of options in the event the stock price goes significantly below the strike and also issue new ESO grants (termed as “reload”) on exercises. Sometimes these features are explicitly embedded in the ESOs and can impact exercise related decisions. To retain tractability, endogenous exercise models often retain focus on one or few of the several possible factors that can influence exercise decisions. For example, for Constant Relative Risk Aversion (CRRA) utilities, Ingersoll [73] derives the subjec-tive and objecsubjec-tive value of ESOs when the employee is constrained to hold a certain fixed proportion of her wealth in the stock of the employing company. Detemple and Sundaresan [49] analyze the value of a non-tradeable option using dynamic program-ming methods under a binomial stock price model for CRRA utilities - that is directly applicable to pricing an ESO. Using a simple 2-period binomial model Kulatilaka and Marcus [82] show how liquidity constraints and other idiosyncratic factors related to an employee can influence the exercise behavior. The authors remark that FASB recommended methods miss out on or inaccurately estimate the effect of such factors. The second approach is to model exercise behavior as an exogenous process. A jus-tification for this is provided by Carpenter [29] who showed that empirically calibrated utility based models do a no better job of predicting exercises when compared to a model that uses random exogenous exercises and forfeits. This motivated researchers to look at intensity based models where exercise is modeled as an independent random process. An example is the model in Carr and Linetsy [31], where exercise occurs as an arrival in a Poisson process whose intensity is albeit modulated by the stock price. This model gives an analytical expression for the ESO cost. In a similar spirit, Hull and White [72], while pointing out drawbacks of the methods proposed by FASB to expense options, suggest an “Enhanced FASB 123” method to expense options. Their approach is to use a binomial stock price process and an employee behavior model

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in which exercise is triggered whenever the stock price hits a certain multiple of the strike. Cvitanic and Zapatero [47] use a similar framework, but in continuous time, which is solvable analytically. They also employ a fictitious barrier based exercise policy for the employee, in this case the employee would exercise her option when the stock price hits a barrier that decreases exponentially with time, and also allow for exercises due to employee exiting the company.

Sircar and Xiong [109] propose an elaborate framework that takes into account reload (wherein exercise of options leads to a new grant) and reset (underwater options have their strikes reset) features of options, and gives analytical formulae for the option price under the assumptions of no expiry and no hedging constraints. Dybvig and Lowenstein in [52], Hemmer, Matsunaga and Shelvin in [66] and Acharya, John and Sundaram in [3] also consider the impact of reload features on option prices. Bodie, Ruffino and Treussard [21] propose a broad framework that an employee can use to weigh ESO benefits while making career related decisions.

2.1.3 Contributions

Most of the proposed ESO costing methods implicitly assume that employees exercise ESOs in an “all or none” fashion. While this assumption is appropriate for traded options, the possibility of partial exercise must be considered while valuing ESOs, as it is reasonable to expect employees to exercise options in batches to distribute the risk over time. Notable exceptions that consider the effect of partial exercises of options are Jain and Subramnaian [75] and Grasselli [61] which provide an analysis of how allowing for partial exercises can impact values and cost of ESOs using primarily a two-period binomial model. Grasselli [61] also considers the possibility of partial hedging using correlated instruments on option prices. Recent work by Leung and Sircar [86] and Rogers and Scheinkman [104] have also considered these effects and solved for optimal exercises numerically using a utility based framework.

What we propose here is to take this reasoning a step further. ESOs are granted in lots and batches and most employees at any given point of time will, in fact, have a basket of unexercised ESOs with varying strikes, expiries and vesting dates. Since, most researchers agree that risk-averseness and over-exposure to the employ-ing company’s fortunes drives early exercise of the ESOs, the degree of this exposure, accumulated primarily through the employee’s own ESO portfolio, will weigh on her decision to exercise an option. Moreover, unlike many of the other quantities al-luded to in ESO pricing models, unexercised ESO grants to an employee constitute

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information that should be readily available to the company and hence easy to use. We therefore argue that unlike the usual approach taken in literature so far to price ESOs, and unlike FASB recommended methods, ESOs perhaps need to be priced not one by one but as an entire portfolio of options held by a particular employee. Thus, ESO exercises can not only be “partial’ but also “coupled”. This would also mean that ESO costs in general will not be linear. For example, the cost of a lot of ESOs does not increase linearly with its size. More generally speaking, the cost of an ESO portfolio will not be the same as the sum of its parts.

In this chapter, we examine the case for portfolio pricing of options by trying to study the qualitative implications of a portfolio approach on ESO costing. We use a standard expected utility maximizing framework as a basis for the employee’s decision process. We begin with the case where employees have several ESOs with the same terms. We find that even in this simple case, exercise policies and hence the implied cost of the grant for the employing company can, in general, vary arbitrarily depending on the nature of employee’s utility function and the stock price dynamics. However, for commonly used utility functions such as the class of Constant Relative Risk Aversion (CRRA) and Constant Absolute Risk Aversion (CARA) utilities, for any stock price dynamics diversification needs would cause the employee to exercise a proportionally larger component of her grant earlier. As a result, the cost of the portfolio in these cases increases sub-linearly with the grant size for options with similar terms. We find this effect interesting because it suggests that ESOs not only have diminishing marginal utility for employees, as one would expect, but in some sense also diminishing marginal costs (or more generally, diminishing average costs) for employers issuing them. We then seek to extrapolate these findings to the case where the employee has multiple types of ESOs in their portfolio. Surprisingly, even for CARA utilities, cost of an option portfolio can turn out to be super-additive, i.e., more than the sum of it parts for some stock price dynamics. However, with an additional but reasonable assumption on stock price dynamics that can be linked to diversification, the cost of the portfolio with multiple types of ESOs can be shown to be sub-additive or less than the sum of its parts. Our analysis thus establishes that a one-by-one costing of ESOs is likely over-estimating the cost of ESOs.

2.1.4 Chapter Layout

In Section 2.2, we briefly describe the model used in this chapter. In Section 2.3, we discuss the simple two period case and conditions on utility functions that will

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make the average cost of an option grant decreasing in the size of the grant. Next, in Section 2.4, we consider the multi-period case for a single option type and show that the average cost under both CARA and CRRA utilities will be decreasing for arbitrary stock price dynamics. In Section 2.5, we then consider the case of ESO portfolios with multiple option types and show that there exists a partial order of exercise between options of different types. In Section 2.6, we examine in detail the problem of expensing multi-type multi-period option portfolios, particularly for CARA utilities. Section 2.7 provides a summary of the results obtained.

2.2 Model

We work with a discrete time model, where the employee treats her ESO grants as investment rather than consumption instruments. Further, the employee will never require to exercise her ESOs for liquidity reasons. We do not consider the impact of the employee quitting or being fired on option exercises or costs. We also do not take into account the “reset” and “reload” tendencies/policies that companies sometimes have in more exotic ESO grants as we would like to retain focus on the key goal of this chapter i.e., the impact of a portfolio approach on exercise behavior and the implied option cost.

The employee has a concave utility function U (·) and a planning horizon T and seeks to maximize the expected utility of her wealth position WT at T .

In our model, the employee may have N different type of options in her portfolio P. The type i option is characterized by a strike price, denoted by Ki, an expiry Ti

and a vesting date Vi. Also, the number of unexercised options of type i in P at time

t are denoted by αi,t, with αi 4

= αi,0. Restricted stock grants can be treated within

this framework as options with strike 0. We also impose the natural restrictions that the employee can neither trade nor hedge against these options. We allow for partial exercises and for mathematical convenience “fractional exercises” to avoid integrality constraints.

We assume that all non-ESO wealth is invested in a well diversified portfolio, whose returns are independent of the employing company’s stock. Any proceeds from option exercise are also likewise invested. An assumption that helps us simplify the analysis considerably is that the employee continues to measures her wealth at time T in units of wealth indexed to some reference time, by discounting time t cash-flows a subjective discounting factor βt. βt can be interpreted as an “opportunity

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cost of cash” for the employee5_.

This approach is in similar spirit but slightly more general than the one used in several utility based models for exercising ESOs, notably Kadam [77], Huddart and Lang [70], Kulatilaka and Marcus [82] and more recently, the model discussed in Rogers and Scheinkman [104]. In principle, upon exercise, the employee has the freedom to invest the proceeds along with other non-option wealth in the markets, and hence must jointly solve the problem of investing non-option wealth and exercising ESOs. Such an approach has been taken for example in Leung and Sircar [86], Grasselli [61]. Allowing for this additional flexibility usually requires making some other restrictive assumptions in order to maintain analytical tractability. Both Leung and Sircar [86] and Grasselli [61] assume that the employee has Constant Absolute Risk Aversion (CARA) type utilities and also assume simple dynamics for stock price processes. By using a model where the employee always uses her wealth at time 0 as a numeraire and subjectively discounts future cash-flows, we can decouple the exercise decision from the non-option wealth investment decision and simplify the analysis considerably. We, in fact, assume no particular form of dynamics on the stock price process except that it is a Markov process, to keep notation less cumbersome.

Under the model described above then, the employee’s decision problem can be described by the following optimization problem. βt denotes the time t discount

factor for the employee, based on her opportunity cost of cash and W , the current non-option wealth in the reference time units.

max V _{= E[U (W}T)] ; s.t. WT = W + N X i=1 T X t=s xi,tβt(St− Ki)+ , xi,t is Ft− measurable. , T X t=0 xi,t = αi. . . 1 ≤ i ≤ N , xi,t = 0 if t < Vi or t > Ti . (2.1)

The exercise problem in (2.1) is a dynamic programming problem.

The quantity xi,t denotes the number of options of type i to be exercised at time

t. The expectations in (2.1) are with respect to the employee’s belief about the

5_{The analysis presented carries through even when the subjective discount factor β}

tis taken to

be stochastic. We only require that βtis almost surely decreasing with time. For simplicity, we will

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employing company’s stock price process. We will assume that there exists a unique preferred solution to the problem in (2.1). To make this precise, if there are multiple solutions to the problem, then the employee chooses for implementation at any given time an exercise policy that has the smaller value of the ordered set {xi,t : 1 ≤ i ≤ N },

when comparisons are made in the lexicographic order. We denote the exercise policy so obtained by x∗ and the optimal exercise for option i at time t by x∗_i,t.

We assume, there is also a unique risk-neutral measure, Q, that prices securities, and is absolutely continuous with respect to the real or believed stock price process. Let Dt= exp(−

Rt

0 rsds) denote the time t discount factor based on risk-free interest

rates rs. Then the cost of the grant to the employer is given by:

C(P ) = EQ " _N X n=1 T X t=0 x∗_i,tDt(St− Ki)+ # . (2.2)

We also assume that the stock does not pay dividends. The quantity Dt(S0− K)+ is

a Q sub-martingale by Jensen’s inequality.

Most people, typically expect stocks to appreciate on average over time, i.e., be-lieve the stock price process to be a sub-martingale. By Jensen’s inequality, the expected value of an option’s payoff which is a convex function of the stock price, is increasing in the time of exercise. However, the employee’s utility function is con-cave and hence delaying exercise need not increase the expected utility of the payoffs received for the employee. Thus, there is a trade-off between exercising immediately and waiting, and the exercise decision will be impacted by several factors including the nature of utility function, the time to expiry, assumed dynamics of stock prices, the level of current non-option wealth and the number of unexercised options.

Our goal in this chapter is to analyze qualitatively the nature of an employee’s exercise policy, as governed by (2.1) and thereby get some comparative statics on the cost of issuance to the employer, as given by (2.2). We start with a simple case, where N = 1, i.e., the employee has several options of a single type in the portfolio.

2.3 Nature of ESO Cost Functions

In this section, we show that even for concave utility functions, the cost of issuing ESOs can become convex.

Let us take the simplest instance of (2.1), with the number of types N = 1 and expiry time T = 2.

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We will assume that the options in question have already vested. The correspond-ing exercise decision problem is then,

max V _{= E[U (W + xβ}0(S0− K)++ β1(α − x)(S1− K)+)] ;

s.t. 0 ≤ x ≤ α . (2.3)

As there is only one type of option involved, we have dropped the sub-scripts ref-erencing the option type. The quantity x denotes the only decision variable in this problem - the number of options to be exercised at time 0. If the optimal value of this decision variable is x∗, then the implied cost to the employing company is given by

C(α, W0) = x∗(S0− K)++ D1(S1− K)+ .

We find that, even for this simple case, the employee exercise policy or the implied cost for the employers cannot be generalized. The value of the option grant to the employee, as measured in terms of her optimal attainable utility will always be concave in the option grant, so long as the employees utility function is concave. As a function of the grant-size, the cost can however become super-linear.

We now give a simple example where the cost of the option grant becomes convex.

Example 2.1. The employee has a utility function

U (y) = min

y,1 3y + 40

and an initial wealth W = 50. The employee’s utility function is concave (though not differentiable). Assume the employee also has α ESOs with strike 90 expiring in one period, i.e., T = 1. To keep things simple, we set both the opportunity cost of cash as well as risk-free interest rates to zero. Suppose the current stock price S0 = 100

and the employee believes that at T = 1, S1 will be either 120 or 80, with equal

probability. The probability of these movements in the risk neutral measure can then also be verified to be 1

2. This implies that the fair value of an American call option

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of options, x∗₀, that the employee should exercise at time 0 is given by x∗₀ = min(1, α).

Thus the resulting cost C(α, W ) to the company is:

C(α, W ) = 10 · min(1, α) + 15 · (α − 1)+ (2.4)

(2.4) shows that the cost of this grant to the employing company is convex in the size of the grant in this case.

In Lemma 2.1 that follows, we characterize a class of utility functions for which the cost function becomes sub-linear in grant-size. This shows that mere concavity of the utility function, which is sufficient to infer diminishing marginal value to the employee, is however not sufficient to draw any conclusions about the implied cost functions.

Lemma 2.1. Consider the two-period, single ESO type, exercise problem in (2.3) and the corresponding cost function (2.4). We assume that the employee’s utility function U (·) is twice continuously differentiable. Then, the average cost of an ESO i.e., C(α,W )_α is decreasing in α, for all values of W > 0, irrespective of the believed dynamics of S1, if the following condition on the utility function U (·) is satisfied:

−y·U00_(x+y)

U0_(x+y) is increasing in y for all x > 0, y > 0.

Proof. We fix the non-option wealth W and treat it as a constant for this proof. We first rewrite the problem (2.3) in terms of a different control variable y =4 _αx. Also for ease of notation we let P0

4 = β0(S0− K)+, P1 4 = β1(S1− K)+ and ∆ 4 = P1− P0. Then

the employee’s problem is

max

0≤y≤1V = E[U (W + αyP0+ α(1 − y)P1)]

= E[U (W + α(P1− y∆))]. (2.5)

Let us first assume that the optimal y in (2.5), say y∗, satisfies 0 < y∗ < 1. Then (2.5) is a concave maximization problem. First order optimality conditions require

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Differentiating (2.6) w.r.t. α, we get E U00(W + α(P1− y∗∆)) ∆ P1− y∗∆ + α ∂y∗ ∂α∆ = 0 ; i.e., αE−U00_{(W + α(P}

1− y∗∆))∆2

∂ y

∗

∂α

= E[−U00(W + α(P1− y∗∆))(P1− y∗∆)∆] . (2.7)

Now, if the condition specified in the lemma is satisfied, as W > 0 and P1− y∗∆ =

y∗P0+ (1 − y∗)P1 ≥ 0, we must have −U00_{(W + α(P} 1 − y∗∆))α(P1− y∗∆) U0_{(W + α(P} 1− y∗∆)) ≥ αγ(W + αP0); if ∆ > 0 , −U00_{(W + α(P} 1 − y∗∆))α(P1− y∗∆) U0_{(W + α(P} 1− y∗∆)) ≤ αγ(W + αP0); if ∆ < 0 ; where γ(W + αP0) 4 = −U 00_{(W + αP} 0)P0 U0_{(W + α(P} 1− y∗∆)) .

Hence, we must have

−U00(W + α(P1− y∗∆))(P1 − y∗∆)∆ ≥ γ(W + αP0)U0(W + α(P1− y∗∆))∆ .

Using this fact in (2.7), we get

αE−U00(W + α(P1− y∗∆))∆2 ∂ y ∗ ∂α ≥ γ(W + αP0)E[−U 0 (W + α(P1− y∗∆))∆] = 0 .

Now as U (·) is concave, U00(·)∆2 < 0. This in turn implies ∂y∗ ∂α ≥ 0 . But, C(α, W ) α = C1− y ∗ (C1− C0) ,

where C1 is the fair value of the European call option with strike K maturing at

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that the average cost of a grant, i.e., C(α,W )_α is decreasing in α in the region where 0 < y∗ < 1. Now consider an ˆα, such that y∗ = 0. Since y∗ must be continuous as an implicit function of α, if y∗ > 0 for some α, then we must have y∗ = 0 for all α < ˆα. Similarly, if y∗ = 1 ⇒ C( ¯α,W )_α_¯ = C0, for some ¯α, then y∗ = 1 for any α > ¯α.

As C0 ≤ C1, this completes the proof.

Remark 2.1. The condition in Lemma 2.1 is satisfied by the two most commonly used classes of utility functions. For CARA or exponential type utility functions, where U (x) = −e−cx with c > 0,

−U00_{(W + y)y}

U0_{(W + y)} = cy ,

which is indeed increasing in y. For CRRA, or power utility functions, where U (x) = x1−a_1−a−1, a ≥ 1, we have

−U00(W + y)y U0_{(W + y)} = a y W + y = a 1 − W W + y ,

which is again increasing in y.

Remark 2.2. Note that Lemma 2.1, shows that the average cost of an ESO grant is decreasing in the size of the grant for certain utility functions. This is a weaker condition than to say that the cost of the ESO grant is concave. The latter implies decreasing marginal cost and consequently subsumes decreasing average costs.

In the next section, we examine the nature of ESO cost functions for CARA and CRRA utilities for the case where the employee’s ESO portfolio consists of options of a single type, but expiring after several periods.

2.4 The multi-period problem for one type of

op-tions and nature of cost function

In this section, we seek to characterize the ESO cost function, as in Lemma 2.1 for the multi-period case. We continue to assume that the employee has only one type of options, i.e., N = 1. Dropping the sub-scripts corresponding to the option type,

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the multi-period version of the single ESO type portfolio can be written as max V _{= E} " U W + T X t=0 βtxt(St− K)+ !# ; s.t. T X t=0 xt = α , xt ≥ 0 . (2.8)

As before, we assume that there is a unique preferred optimal exercise policy6 _denoted

by x∗. The corresponding cost function (2.4) is given by

C(α, W0) = T

X

t=0

Dtx∗t(St− K)+ .

We next consider the CARA and CRRA utility cases separately and show that as in the two-period case, the average ESO cost is decreasing in the size of the grant even when the expiry is several periods away.

Exponential or CARA Utilities

The function U (·) in (2.8) in this case is given by U (y) = −e−cy, for some c > 0. We first show that the optimal exercise policy for this class of utility functions assumes a relatively simple form. In fact, it is independent of the non-option wealth level W .

Lemma 2.2. For CARA utilities, the optimal exercise policy x∗₀ is independent of the employee’s non-option wealth level W and has the form x∗₀ = (α − η∗)+, where η∗ is a quantity that is independent of α, and W .

Proof. Suppose an exercise policy, x∗ maximizes (2.8) for some α and W > 0 for the CARA utility U (y) = −exp(−cy). This happens if and only if x∗ is a solution to the

6_{In case of multiple competing optimal policies, the employee chooses the one that requires}

exercising the fewest number of options immediately. This policy should lead to a conservative estimate of costs to the employer.

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problem min V _{= E} " exp −c T X t=0 βtxt(St− K)+ !# ; s.t. T X t=0 xt = α , xt ≥ 0 .

which has no dependence on W . It then follows that the optimal exercise policy is independent of W .

Since we are only concerned with the dependence of exercise policy on α and W , we fix the other parameters, i.e., current stock price S0 and the discount factor β0

that should be applied to any payout received in current period to convert it to its reference time equivalent and treat them as constants. The optimal number of options to be exercised in the current period can be characterized as

x∗₀(α, W ) = inf{x | 0 ≤ x ≤ α and x∗₀(α − x, W + β0(S0− K)+) = 0} .

Using the independence of optimal policy from non-option wealth, then we have x∗₀(α, W ) = inf{x | 0 ≤ x ≤ α and x∗₀(α − x, 0) = 0} . (2.9) Now, we define the set A0 and η∗ as follows:

A0 4

= {α ≥ 0 | x∗₀(α, 0) = 0}

η∗ = sup A4 0 = sup{α > 0 | x∗0(α, 0) = 0} (2.10)

Suppose A0 is empty, then we simply set η∗ = ∞ and the lemma holds, as x∗0(α, W ) =

0 for all α, W . The proof is also trivial if x∗₀(α, W ) = α for all α, in which case we set η∗ = 0. Hence, we consider the case that A0is not empty, i.e., x∗0(α, W ) = x∗0(α, 0) > 0

for some α > 0. We claim that in this case

• A0 is bounded above and η∗ is finite

• If α < η∗ _{then x}∗

0(α, 0) = 0

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exist α, ¯α such that α < ¯α and

x∗₀(α, 0) = 0 and x∗₀( ¯α, 0) = 0

but x∗₀(α, 0) > 0 if α < α < ¯α .

Let x∗₀( ¯α − δ, 0) = y for some δ : ¯α − α > δ > 0. Then, using (2.9), y = α − α − δ .¯

But, this would mean

lim

α→ ¯αx ∗

0(α, 0) = ¯α − α > 0 ,

while x∗₀( ¯α, 0) = 0. The optimal exercise policy is then discontinuous in grant size, which leads to a contradiction.

Hence η∗ is finite and

x∗₀(α, 0) = 0 if α < η∗ . (2.11)

It then follows from (2.9) and (2.11) that the optimal exercise policy is given by x∗₀(α, W ) = x∗₀(α, 0) = (α − η∗)+ ,

which is of the desired form.

Corollary 2.1. For CARA utilities, the average cost of an ESO grant is decreasing in the size of the grant.

Proof. We prove this by induction on the number of time periods to expiry. We can actually prove a stronger result - the marginal cost of issuing an ESO in the case of CARA utilities is decreasing, i.e., the cost function is concave in grant size. The result holds trivially for T = 0, i.e., when the option is expiring immediately. Suppose it holds for T = m. Let C(α, W, m) denote the cost of α ESO grants with m periods to go, when the employee has non-option wealth W . (We fix the current price of the underlying to S0 and the accumulated discount factor is β0.) We have

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• x∗

0 = 0: Then C(α, W, m + 1) = EQ[D1C(α, W, m)|F1], and the concavity of

cost property follows from induction hypothesis. • x∗

0 > 0: Then, from Lemma 2.2 it follows that the marginal cost of the ESO

grant is (S0 − K)+ for all α > η∗, with η∗ as defined in Lemma 2.9. Since the

stock does not pay dividends, by Jensen’s inequality the discounted European call option payoff, i.e., Dt(St− K)+ is a sub-martingale under the risk neutral

measure Q. Then the marginal cost (S0− K)+for grant size α > η∗ is less than

that for any size α0 < η∗_{, which is at least E}Q_[D

1(S1− K)+|F1].

Remark 2.3. For the exponential utility model, the asymptotic marginal cost of grant-ing an in-the-money ESO, as the size of the total grant α → ∞, turns out to be equal to the options intrinsic value, i.e., (S0 − K). This is the same as the cost at which

companies used to expense ESOs prior to the FASB stipulations in 2005. However, the intrinsic value for this model comes out as a marginal cost and not as the average cost, as was used for cost accounting.

Power or CRRA Utilities

For CRRA or power utilities7 _{U (·) in (2.8) takes the form U (x) =} x1−γ

1−γ, for some

γ > 1. In Section 2.4, for CARA utilities, the optimal exercise policy was shown to be independent of non-option wealth W and to assume a very simple form. For CRRA utilities, the non-option wealth W will impact exercise policy. Nevertheless, as we show in the Lemma 2.3, the dependence of optimal exercise policy in grant size and non-option wealth level is easily characterized in this case as well.

Lemma 2.3. For CRRA utilities, the optimal exercise policy has the form

x∗₀(α, W ) = θα − (1 − θ) W β0(S0− K)+ + ,

where θ is independent of α, W and 0 ≤ θ < 1. S0 denotes the current stock price

and β0 the accumulated discount factor.

Proof. For this proof, we again fix β, the subjective discount factor for current payoffs and the current stock price S.

7_{This class also includes log utilities, i.e., U (x) = ln(x), which can also be considered as a power}

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We first note that the value function is homogeneous for CRRA utilities. If Vd(x, α, W ) denotes the utility derived by following an exercise policy x given an

initial grant α and non-option wealth level W , then we must have Vd(M x, M α, M W ) = M1−γ· Vd(x, α, W0)

Also note that, if xx is a feasible exercise policy to follow for grant α, then M x must be a feasible policy for the grant M α. This means that if x∗ is an optimal exercise policy for a grant of size α and when non-option wealth is W_M, then M · x∗, is the optimal exercise policy for a grant of size α and non-option wealth W . Hence

x∗₀(M α, W0) = M · x∗0 α,W M . (2.12)

Suppose the options have strike K and expiry T . The optimal exercise quantity x∗₀(α, W ) must satisfy the following condition:

x∗₀(α, W ) = inf{x | 0 ≤ x ≤ α and x∗₀(α − x, W + x · β0(S0− K)+) = 0} .

Then, using (2.12) we get

x∗₀(α, W ) = inf x | 0 ≤ x ≤ α and x∗₀ α − x W + x · β0(S0− K)+ , 1 = 0 . (2.13)

Suppose x∗₀ = 0 for all combinations of α and W . Then the form specified in the lemma trivially applies with θ = 0. If not, then we must have S > K. Consider then a certain combination (α, W ), say (αA, WA) such that

x∗₀(αA, WA) 4 = x∗_A> 0 Using (2.13), then x∗₀ αA− xA WA+ xAβ0(S0− K) , 1 = 0 (2.14)

Now, we define the set A0 and quantity κ as follows:

A0 4

= {α ≥ 0 | x∗₀(α, 1) = 0}

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(2.14) shows that the set A0 is non-empty. We now show that

• κ must be finite, i.e., A0 is bounded above and

• If α < κ, then x∗

0(α, 1) = 0.

If either of these claims is not true then as the optimal exercise policy must be continuous in grant size, there would exist α, ¯α such that α < ¯α and

x∗₀(α, 1) = 0 and x∗₀( ¯α, 1) = 0

but x∗₀(α, 1) > 0 if α < α < ¯α .

Let x∗₀( ¯α − δ, 1) = y for some δ : ¯α − α > δ > 0. Then, using (2.13) ¯ α − δ − y 1 + yβ0(S0− K) = α , i.e., y = α − δ − α¯ 1 + αβ0(S0 − K) .

But, this would mean that lim α→ ¯α+x ∗ 0(α, 1) = ¯ α − α 1 + αβ0(S0− K) > 0 ,

while x∗₀( ¯α, 1) = 0; i.e., the optimal exercise policy is discontinuous in grant size, which cannot be the case as the value function is continuously differentiable. From (2.13) and (2.15), it follows that if _Wα ≥ κ, then

κ = α − x ∗ 0(α, W ) WA+ x∗0(α, W ) · β0(S0− K) , (2.16) i.e., x∗₀(α, W ) = 1 κβ0(S0− K) + 1 α − κβ0(S0− K) κβ0(S0− K) + 1 WA β0(S0− K) . (2.17)

Since x∗₀(α, 1) = 0 if α < κ, using (2.12), we conclude x∗₀(α, W ) = 0 if α

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Combining (2.17) and (2.18), we get x∗₀(α, W ) = α − κ · W 1 + κ · β0(S0− K)+ + .

which is of the form stated in the lemma, and the parameter θ = _1+κβ 1

0(S0−K) is

independent of α and W . Note that the exercise quantity given by (2.19) always satisfies the constraint x∗₀ < α.

Lemma 2.3 can be used to show that average costs of an ESO portfolio with a single type of options is decreasing when the employee has CRRA utility. We first prove the following useful result, which formalizes the notion that early exercises tend to reduce ESO costs. For this, the following definition will be helpful:

Definition 2.1. Consider two strategies xA and xB for exercising an option grant of size α. Strategy xA is said to dominate strategy xB if it always leaves a greater number of options unexercised i.e.,

α − t X t=0 xA_t ≥ α − t X t=0 xB_t , 0 ≤ t ≤ T

Next, we show that the cost of a grant associated with a dominant strategy is always higher.

Lemma 2.4. If strategy xA _{to exercise an option grant of size α dominates another}

strategy xB_{, then the option cost C}A _{associated with strategy x}A _{is higher than the}

cost CB _{associated with x}B_.

Proof. For 0 ≤ t ≤ T − 1, recall

αA_t = α −4 t X s=0 xA_s ; αB_t = α −4 t X s=0 xB_s . αA

t and αBt are Ft measureable. As xA dominates xB, we must have αAt ≥ αBt .

Also, let Pt 4

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dividends. Now, CA− CB = EQ "_{T −1} X t=0 (xA_t − xB t )Pt+ (αAT −1− α B T −1)PT # = EQ "_{T −1} X t=0 (xA_t − xB t )Pt+ (αAT −1− α B T −1)EQ[PT|FT −1] # ≥ EQ "_{T −1} X t=0 (xA_t − xB t )Pt+ (αAT −1− αT −1B )PT −1 # (2.19) = EQ "_{T −2} X t=0 (xA_t − xB_t )Pt+ (αAT −2− α B T −2)PT −1 # (2.20) . . . = 0 .

In (2.19), we used the fact that αA

T −1 ≥ αBT −1 and that Pt is a sub-martingale. In

(2.20), we substituted αA

T −1 = αAT −2− xAT −1 and αBT −1 = αBT −2− xBT −1. Note that the

inequality is strict if the discounted option payoff process Dt(St− K)+ is a strict

sub-martingale and there is a non-zero probability of a positive difference in unexercised option positions.

Remark 2.4. Option payoff process is guaranteed to be a Q sub-martingale if the stock does not pay dividends. Lemma 2.4 also holds under a weaker condition - the employee must exercise all her options, whenever the stock reaches a level at which the payoff process no longer remains a sub-martingale under the Q measure.

We now use Lemmas 2.3 and 2.4 to show that ESO cost is sub-linear in grant size for CRRA utilities.

Corollary 2.2. If the employee has CRRA utility, then the average cost of a batch of ESOs, all with the same terms, is decreasing in the size of the grant.

Proof. Fixing, S0, T and β0, from the homogeneity of CRRA utilities, (2.12), it follows

that C(α, W ) = αC 1,W α .

Thus, to prove that average cost is decreasing in α, it suffices to show that cost of the ESO grant is increasing in initial wealth i.e., W . For this, we appeal to Lemma 2.4 and demonstrate that the number of unexercised options associated with a higher

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non-option wealth level dominates the number of unexercised options with a lower non-option wealth level on a path-by-path basis using finite induction.

Consider, for an option with strike K and expiry T , two different combinations of grant sizes and non-option wealth (α0, W0) and ( ¯α0, ¯W0) at time 0, such that α0 ≥ ¯α0

and W0 ≥ ¯W0. We will show that these inequalities are preserved throughout the

option’s life-time.

Suppose S0 ≤ K, then there are no exercises at t = 0, and the first combination

will continue to dominate the second at t = 1. If S0 > K, then using Lemma 2.3, the

difference in unexercised options after exercises at time 0 are accounted for will be

α1− ¯α1 = α0 − θα0− (1 − θ) W0 β0(S0− K) + − α¯0− θ ¯α0− (1 − θ) ¯ W0 β0(S0 − K) +! ≥ α0 − ¯α0− θ(α0− ¯α0) − (1 − θ) W0− ¯W0 β0(S0− K) + ≥ (1 − θ)(α0− ¯α0) ≥ 0 .

Also, the difference in subjectively discounted non-option wealth at t = 1 will be

W1− ¯W1 = W0+ β0 θα0− (1 − θ) W0 β0(S0− K) + (S0− K) − W0+ β0 θα0− (1 − θ) ¯ W0 β0(S0− K) + (S0− K) ! ≥ W0− ¯W0− −θ(α0− ¯α1) + (1 − θ) W0− ¯W0 β0(S0− K) + β0(S0− K) ≥ W0− ¯W0− (1 − θ) W0− ¯W0 β0(S0− K) + β0(S0− K) ≥ θ(W0− ¯W0) ≥ 0 .

Thus the first combination will always dominate the second at the beginning of the period t = 1. By repeating this argument, we see that number of unexercised option associated with the first grant will always dominate the ones associated with the second. Moreover, the difference will become strict whenever there is an exercise. Using Lemma 2.4 then we conclude that when discounted option payoff process is a Q sub-martingale, the cost function is increasing in the non-option wealth position. This, in turn implies that the average cost is decreasing in the size of the grant.

Analysis of employee stock options and guaranteed withdrawal benefits for life

Analysis of Employee Stock Options and

Guaranteed Withdrawal Benefits for Life

by

Premal Shah

B. Tech. & M. Tech., Electrical Engineering (2003),

Indian Institute of Technology (IIT) Bombay, Mumbai

Submitted to the Sloan School of Management

in partial fulfillment of the requirements for the degree of

Doctor of Philosophy in Operations Research

at the

MASSACHUSETTS INSTITUTE OF TECHNOLOGY

September 2008

c

Massachusetts Institute of Technology 2008. All rights reserved.

Author . . . .

Sloan School of Management

July 03, 2008

Certified by . . . .

Dimitris J. Bertsimas

Boeing Professor of Operations Research

Thesis Supervisor

Accepted by . . . .

Cynthia Barnhart

Professor

Co-director, Operations Research Center

Analysis of Employee Stock Options and Guaranteed

Withdrawal Benefits for Life

by

Premal Shah

B. Tech. & M. Tech., Electrical Engineering (2003),

Indian Institute of Technology (IIT) Bombay, Mumbai

Abstract

Acknowledgments

Contents

List of Figures

List of Tables

Chapter 1

Introduction

1.1

Motivation

1.2

Contributions

Modeling Exercise Behavior for pricing Employee Stock

Op-tions

Pricing Guaranteed Withdrawal Benefits for Life

Dynamic Consistency For Sequential Decision Problems

1.3

Thesis Overview

Chapter 2

Pricing Employee Stock Options

2.1

Introduction

2.1.1

Motivation

2.1.2

Related Work

2.1.3

Contributions

2.1.4

Chapter Layout

2.2

Model

2.3

Nature of ESO Cost Functions

2.4

The multi-period problem for one type of

op-tions and nature of cost function